EXCHANGE 


SECULAR  PERTURBATIONS 


ARISING  FROM  THE 


ACTION  OF  JUPITER  ON  MARS 


A  THESIS 

PRESENTED  TO  THE  FACULTY  OP  PHILOSOPHY  OF  THE  UNIVERSITY 

OF  PENNSYLVANIA 

BY 
ARTHUR  BERTRAM   TURNER 

IN  PARTIAL  FULFILMENT  OF  THE  REQUIREMENTS  FOR  THE  DEGREE 
DOCTOR  OF  PHILOSOPHY 


PHILADELPHIA 
1902 


SECULAR  PERTURBATIONS 

ARISING  FROM  THE 

ACTION  OF  JUPITER  ON  MARS 


A  THESIS 

PRESENTED  TO  THE  FACULTY  OF  PHILOSOPHY  or  THE  UNIVERSITY 

OF  PENNSYLVANIA 


ARTHUR  BERTRAM   TURNER 


IN  PARTIAL  FULFILMENT  OF  THE  KEQUIREMENTS  FOB  THE  DEGREE 
DOCTOR  OF  PHILOSOPHY 


PHILADELPHIA 
1902 


EXCHANGE 


PRESS  OF 

THE  NEW  ERA  PRINTING  COMPANX, 
LANCASTER,  PA. 


55 

* 


ACKNOWLEDGMENT. 

I  wish  to  thank  Professor  Charles  L.  Doolittle  and  Mr.  Eric 
Doolittle  for  their  generous  instruction,  and  their  helpful  advice 
in  the  prosecution  of  this  Thesis. 


CONTENTS. 

I.  Lagrange's  Generalized  Equations  of  Motion.     Lagrange's 
Canonical  Equations  ........     5 

II.  Canonical  Forms  of  Hamilton 8 

III.  Method  of  Jacobi  and   its  Application  to  Two  Bodies.     Ca- 

nonical Constants        .         . 11 

IV.  Variation  of  the  Canonical  Constants  and  Jacobi' s  Equation    .  17 
V.  Differentiation  of  the  Equations   containing  the   Canonical 

Constants .  20 

VI.  Transformation  of  the  Equations  Expressing  the  Perturba- 
tions, and  the  Values  of  the  Variations       .         .         .         .23 
VII.  Dr.  G.  W.  Hill's  First  Modification  of  Gauss's  Method  .         .  27 
VIII.  Computation- Action  of  Jupiter  on  Mars        .         .         .         .30 
Biographical  ..........  36 


I. 

LAGRANGE'S  GENERALIZED  EQUATIONS  OF  MOTION. 
LAGRANGE'S  CANONICAL  EQUATIONS. 

Let  Fn,  F12,  F13,  ••  -,  Fln  be  the  forces  acting  on  a  unit  of 
mass  rax, 

F2l ,  F2F2,  F2F3,  •  -  •,  F2n  be  the  forces  acting  on  a  unit 
of  mass  m2, 

etc.  etc. 

Let  8pu ,  &pl2 ,  Sp13 ,  •  •  • ,  8plH  be  the  virtual  velocities  of  ml , 
tyPn.  >  ^22  >  ^23 '  ' ' ' »  ^Pa»  ^e  *^e  virtual  velocities  of  77i2 , 

etc.  etc. 

Now  assume  that  each  mass  m.  be  displaced  an  infinitesimal  dis- 
tance I  =  ds.  in  the  direction  in  which  the  mass  m.  would  have 
moved  during  the  next  instant  had  it  not  been  subjected  to  this 
arbitrary  displacement,  and  let  the  distance  in  each  case  be  pre. 
cisely  equal  to  the  distance  which  the  body  would  have  moved  dur- 
ing the  next  instant  had  it  not  been  subjected  to  displacement. 
Then  by  the  theorem  in  virtual  velocities  that  ]£  FSp  —  &t  =  change 
in  the  living  force,  we  shall  have  for  the  masses  ml  •  •  •  mk , 


=  OJL     tor     m, 

mmmm  &  in,      JL     «/»  h 

1 

adding  we  get 
(a) 


1  1 

5 


6 

These  equations  involve  the  masses  because  F^  are  forces  on 
unit  mass. 

Now  it  is  known  that  the  change  in  the  living  force  of  a  system 
is  equal  to  the  work  done  on  the  system  and  since  work  equals  force 
X  distance,  we  shall  get  for  the  change  in  the  living  force 

V    /  f    *       i   sl-i%         i 

1  \AJ\J 

Equating  these  two  values  of  S  T,  we  get, 

=  0 


which  is  Lagrange's  Generalized  Equation. 

If  now  we  suppose  the  forces  to  be  resolved  along  the  three 
coordinate  axes  the  above  equation  can  be  easily  made  to  assume 
the  form, 


where  X,  Y,  Z  are  the  total  components  of  the  forces  along  the 
coordinate  axes. 

Let  us  assume  a  certain  function  U  (Potential  Function)  which 
is  independent  of  the  time  t,  such  that 

W  SU  dU 

->       ^=  -^JL  ,  ~~^i       =    JL   ,  ~       ==   &  , 

dx  dy  dz 

then  by  substitution  equation  (2)  becomes 


ifc  ^  +  '  •  -etc-)  =  2  (mwBx  +'"  etc*)' 

Now  the  left  hand  member  of  this  equation  is  the  total  varia- 
tion of  £7,  or  8V. 

Since  T  (Living  Force)  =  \  mi?,  &T=  mvSv,  but 


1 

d2x  dv 

mW^^mdt 

and  adding 


^  dv 

m  -j-z  ox  =  m  -j-  ox  +  mvov  —  o  T 
at  at 

now 


mv  ^i 


for 
Hence 


(Sx)  =  mvS  I  -5—  \  —  mv8v. 


or 

(3)  SU=jt 

Let  us  suppose  I7  to  be  a  function  of  the  independent  variables 
ql9  q2,  -  -  -,  etc.,  then  the  variation  of  T  is 


etc., 


3s 
&t-3j*9i+  • 

These  values  substituted  in  (3)  give  the  equation 


and  since  the  ^'s  are  independent  we  can  equate  the  like  variations 
and  obtain  the  following  partial  differential  equations  : — 


dJ?—iL( m  ^\__^ 

dql  ~~  dt  \       dql )       dql 


etc.  etc.        etc. 

which  become 

dU      d  / dT\  dT 

(4)  dql  ~  dt\  dq(  )  dql 

etc.  etc.        etc. 

Since 

ds      —-^dsdq      ds  dv       ds 

v  —  —  =  /   —  •  — -  -f-  —     and     — -,  ==  — . 
dt      ^^  do    dt       dt  dq,       dq, 

But  J  mv2  =  T,  therefore 

dT  dv  ds 

—,  =  mv  =-7  =  mv  = —  • 

These  equations  are  known  as  Lagrange's  Canonical  Forms,  and 
in  deriving  them  we  have  assumed  that  all  points  of  the  system 
have  been  expressed  in  terms  of  £,  and  k  independent  variables 
ql  •  •  -  qk .  Since  there  are  &n  coordinates  altogether  in  the  system, 
(icx  yl  zl9  •  •  -,  xn  yn  zn)  this  assumes  that  there  are  (%n  —  k)  equa- 
tions of  condition. 

II. 

CANONICAL  FOKMS  OF  HAMILTON. 

Let  us  still  regard  T  as  expressed  in  terms  of  q ,  •  •  • ,  qh , 
%i •>  '">  Qk)  anc*  wrl*e 

_dT  _dT 

T  was  originally  a  homogeneous  function  in  regard  to 
dxl      dx2 


9 

and  since  dx  ,  dy  ,  dz  ,  •  •  •  are  connected  with  ^J  ,  q'2  ,  •  •  •  by  linear 
equations,  T  regarded  as  a  function  of  q  and  ^'  is  homogeneous 
and  of  the  second  degree  in  q[  ,  ^  »  •  •  •  •  It,  therefore,  satisfies 
Euler's  equation,  or 

dT  dT 


Taking  the  variation  of  T 

2S2'to,,)  = 
and  by  direct  variation 


a? 

subtracting 


but 


Equating  like  variations  we  get 
dTH 


etc. 


_ 

dq   ~  dq 


etc. 


where 

••••,  2 


Now  let  ZT=  jf—  C/",  where  ZT==  constant  independent  of 
then 

dU 


and  equations  (4)  give 

subtracting 

Again 


10 

dU     dPl 
dql  ~     dt 

dt  = 


dH 


Now  U  is  supposed  not  to  contain  q[ ,  pl ,  or  t ,  hence  d  Ujdpl  =  0 , 
and  we  have  just  shown  in  equations  (4)  that 


dT 


dt 


hence  by  substitution 


We  thus  have  the  systems  of  equations — 


(5) 


dt 


dH 


dH 


dt 


dqi       ,  dH 


dt 


dH 


dt 


These  equations  reduce  the  system  of  k  differential  equations  of 
the  second  order  to  2k  differential  equations  of  the  first  order.  If 
we  call  pt  and  q.  conjugate  independent  variables,  Hamilton's 
reduction  may  be  stated  thus :  "  Hamilton's  Canonical  Forms 
arise  from  finding  two  series  of  variables  in  terms  of  which  the 
coordinates  x ,  y ,  z ,  can  be  expressed.  The  total  differential  of  any 
one  variable  with  regard  to  the  time  is  equal  numerically  to  the 
partial  derivative  of  a  certain  determinate  function,  H,  with  regard 
to  the  conjugate  variable." 


11 
in. 

METHOD  OF  JACOBI  AND  ITS  APPLICATION  TO  Two  BODIES. 

CANONICAL  CONSTANTS. 

Let  us  suppose  these  2k  equations  of  Hamilton  to  be  integrated, 
then  we  will  get  2  Ic  constants  of  integration  cx  •  •  •  ca,  and  let  us 
take  the  partial  of  ZTwith  respect  to  c,  ,  since  H  will  be  a  function 
of  the  c's,  whence 


_  ±  dp 

~~  ~ 


and  by  substituting  from  (5)  this  becomes 


_  ,  \ 

"'i      "'  dci  1      '  / 


,    dp,  \          dp,    dq, 

t'dcl 

d  (      da, 
=  ^A^ 
But  2T=  Z^i(^i)/ 

d        dql 


d  f     dq,  \ 

~M^  +  ''7* 


Now  ZT=  ^T—  C7",  and  substituting,  then, 

)       dfdq 
~P         H 


Integrating  with  respect  to  £,  we  get 

Assume 

and  multiply  by  cZcx,  then 
and  8$-- 


12 


Taking  the  variations  of  S  directly  we  get 


and  since  the  ^'s  are  independent 


(6) 

We  have 

~di==~dt+  £ 


etc. 


-.Pio 


etc. 


dS 


dS 

'    and 


We  here  consider  If  as  a  function  of  the  2&  constants  ^ 

,  but  independent  of  t. 
The  equation  (Jacobi's) 

dS       d$ 


when  integrated  will  give  S  containing  the  k  constants  ql  •  •  -  qk , 
and  since  the  partials  of  S  with  respect  to  these  k  constants  are  to 
be  put  equal  to  k  constants  by  (6),  we  shall  have  introduced  upon 
integrating  this  last  series  of  k  partial  differential  equations  of 
the  first  order,  2k  constants  altogether. 

To  integrate  such  a  differential  equation  of  the  first  order,  we 
have  need  of  Euler's  Transformation,  which  is  derived  as  follows  : — 

Suppose  Z  =  <f>  (ojjO^cCg  •  •  • )  and  we  desire  to  integrate 


0. 


13 

Assume  y  =  z  —  x^  dzjdx^  =  z  —  x^  ,  then 

dy  =  dz  —  x1  dx[  —  x(  dx1  , 
but  from  the  equations  in  »,  we  have 
dz  dz   . 


and  by  substitution 

dx~  dXl  +  ' 

(d*   i  \ 

=  I  -j-  dx~  +  •  •  •  )  —  x.  ax,  . 
\dx2     ^       ) 

Therefore 

dy  dy       dz 

dx(=    ~x^     ^2  =  ^2)'">etC') 
and 

^  =  0  =  — 

dxl  dxl  ' 

These  values  substituted  in  the  original  differential  equation, 
gives 


Our  new  equation  contains  the  same  number  of  variables  as  the 
original  equation,  but  the  variable  xl  is  replaced  by  (  —  dyjdx^)  . 
If  this  latter  is  a  constant  by  the  conditions  of  the  problem  we 
have  thus  removed  a  variable. 

It  is  easily  shown  that  the  equations  of  undistributed  motion 
for  two  bodies  are, 


in  which  problem 


14 

d*z_ 

"3?  " 


r 

TT /77  7"7" 

JJ_    sss    _/     —     (_/    :^ 

>  r 

Now  Jacobi's  equation  is 


and  hence  ^T  must  be  expressed  in  terms  of  the  new  variables.     Let 
dx 

ft-*  «.-s 

?2  =  y,     then     jJ_J,     and     T  =  |m  [(,;)«  +  (?;)2  +  &)*] 

6?» 

?a  =  »  ?,-S 

by  substitution 


or 

i 

Now  transform  to  polar  coordinates  by  means  of  the  equations 
x  =  r  cos  a-  cos  z> 
y  =  r  cos  <r  sin  v 
z  =  r  sin  cr         whence 

2         1       /dxSV     1  /£#V1     >^2. 


da  )  J"  r  ~°' 


15 

Now  apply  Euler's  transformation  by  letting  S'  =  S  -f  cut  ,  then 
dSJdt  =  —  a,  [a  is  a  constant  of  integration]  and  substituting 


Solving  we  get 


again,  let   /S"'  =  »S"  —  a^,   then   dS'jdv=al  and   our   equation 
becomes 


which  can  be  written 


Put  the  left  member  =  a\ ,  then 


r2 
or 


-         and 


Hence  the  complete  integral  gives 


Xr    \           2&2       a2             Cv     \             o> 
-J2a+-- £dr+        -Ja2 2L 
\                r          r"               i       \             COS 
•^o 

Now  ^  =  S'  -  a^,  and  ^  =  S  +  at, 

r  I      2F    of 

.S-^ir      ««  +  J^  ^2^  +  —  -^    r 


a22- 
J0     \   2 


cos 


16 


Put 


then 


a#          as 
i 


and 
dr 


=  +  ^  - 


/^<r  <&T 

V.  7T3 


cos^o- 


cos  cr 


where  rx  is  the  least  root  of  the  equation 


Let  r  =  r1 ,  then  —  t  =  /3 ,  or  —  /3  =  Time  of  Perihelion  Passage, 

£2 

If  PJ  =  smallest  root 


largest      " 


then 


and 


r1_a(l  —  e),  ^^ 

hence  equating  we  get 

2a=  -  — , 
a. 


—  -  e  . 


From  the  value  of  /^  we  see  that  a2  /cos2  a  =  a2  determines  the 
maximum  value  of  cr  (i.  e.,  <7  =  i)  whence 


17 

«1  =  a2  cos  cr  =  k  i/5  i/l  —  e2  cos  i  , 
and  when 

cr=0,     i>=O,     .-.^=11. 

Let  ^.P  (7  be  a  right  spherical  A  in  which  /  PA  C—i^  AP  —  r) 
and  P  C  =  &  ,  then  sin  <r  =  sin  i  sin  77  ,  and  cos  <rd(r  =  sin  i  cos  77^77 
which  substituted  gives 

S*<r  d<T  S*cr  C?<7 

**  I  ~n  ~^r  '**  /    ("TTT  cos2f 

/  \    <*l  --  2~  1  \    al       1    --  2— 

J  o      \    2       cos2  cr  »/  o      \    2  \          cos2  cr 


COS  crjcr 


/      o 
V  cosh 


er 


rjcr  Ft 

^=f^  = 
—  Sln2^      J0 


If  the  body  is  at  perihelion,  then  77  =  co  ,  and  y52  =  co  ,  or 
52  =  TT  —  il  .  Therefore  our  six  constants  of  integration  or 
canonical  constants  have  the  values 


(8) 


*  i/a  1/1  —  e2  cos  *» 


IV. 

VARIATION  OF  THE  CANONICAL  CONSTANTS 

AND  JACOBI'S  EQUATION. 
The  equations  of  motion  for  two  bodies  are  of  the  form 

d2x      dU  u 

-Ta  =  ~^—  •>      where      U  =  -  . 
dt2       dx  r 

When  a  third  body  is  added  to  the  system  U  is  of  the  form 


where  It  is  the  Perturbing  Function.     The  question,  therefore,  is 


18 

to  find  what  change  must  be  made  in  the  canonical  constants  in 
order  to  replace  H  by  (  H  —  If)  in  the  equation  just  solved. 
From  Hamilton's  Forms,  eqts.  (5),  we  have 

dql_dH 

~dt  ~  dj^  ' 

and  if  ff=  H  —  R,  becomes 

dql  _  d(ff-R)  _  dH  _  dR 

dt  ~          dPl         "  dPl  ~  d^ 
(9)     likewise 

^i=    -dj^=      d(H-R)  =   _dH     dR 
dt  '        dqi  '  dqi  Wi+d%i' 

Considering  pl  and  q^  as  functions  of  the  constants  and  £,  we 
regard  the  constants  as  variables  and  find  what  variations  must 
take  place  in  them  so  that  H  may  be  replaced  by  (H—  R),  that 
is,  p^  and  q,  must  satisfy  (9)  . 

Assume 


*,  ^  ---  ft,  0        then 

i    —  4-  ^    ^ 
dt  da  '  dt        d/3     dt 


^  ^a     a^  ^\ 

^  '^        ^  '  dt  ) 


now  equation  (5)  gives 

d^-dH     aid  *' 

—  =  —  ==  -^  -         d/llU  —  _ 

a^       ^Pj  dt 
which  substituted  in  (9)  gives 


'a  dt  Sft  dt 
(10) 

dR      ^  ( dpi  da  dpl  dfi 

a  dt  d/3  dt 


19 

Since  we  can  find  p  and  q  exclusively  in  terms  of  the  a's  and 
/3'Sj  and  also  a  and  fi  in  terms  of  the  />'s  and  q's ,  we  can  apply 
Jacobi's  Theorem  which  states  that, 

(a)         -~-  =  -~  (c)         ~£  —  -~^ 

^ak  ^9.i  ^fik          ^Pi 


whence 

do,  d/3      da,       da       dp.       d  8 

-^rL==  —  jr-'>     -^=5—'     -^  =  ^r- »  and 

da  dpl       op       dpl       da       dql 

and  making  these  substitutions  in  (10) 

dj3    da       da    d/3 


dpl  ~  *- 

d H      —  f  d/3    da        da 


da    df* 
di ' 


If  we  express  R  in  terms  of  the  p's  and  ^'s  then  by  Calculus  of 
Variation,  we  have 


and  substituting  from  above  the  values  of  dR  /dp  and  dltjdq,  we 
get 

da      da    d3  d*    da      da 


which  can  be  written 

3/3  80         da  I8a  da         dfi 


Since  a  and  /3  can  be  expressed  in  terms  of  the  p's  and  #'s  ,  let 
us  assume 


then 


and  by  substitution  in  &??,  we  have 


If  now  J?  is  expressed  in  terms  of  the  a's  and  yS's  ,  then 


and  equating  like  variations,  we  have 


dt 


etc. 


dt 


etc. 


These  are  Jacobi's  equations  and  they  give  the  total  variation  of 
the  constants  in  terms  of  the  partial  of  the  Perturbing  Function, 
when  the  latter  has  been  expressed  as  a  function  of  the  constants 
(a,  /3)  and  the  time  (t). 

V. 

DIFFERENTIATION  OF  THE  EQUATIONS  CONTAINING  THE 

CANONICAL  CONSTANTS. 
Solving  equations  (8)  we  find 


(12) 


cos  i  =  -1 , 


and  let     e  =  TT  —  nr , 
P ,       then       e  =  /^  4-  /32  • 


since      n  =  — = . 
By  differentiation  of  the  first  of  these  equations 


21 


da       1&     da.      2a2  da 
da 


da 
dt 


(13) 


?  J  a/3 ' 

in  like  manner  we  can  get 
de      a  i/l  —  e2 


sin  ij-.  =  -   ,-    /:==    cos  i  ^~ =-3- 

dt      Jc  i/a  i/l  —  e2  L         ^2       3ft 


^n 

^ 

W 

de 
dt 


kdR 

a*  da 


dR 


Since  e  is  the  only  equation  in  (12)  containing  y8,  and  7?  being 
a  function  of  the  a's  and  /3's ,  then 


Likewise 


but 


dR  _dR    de  _k_    dR 

~d£$=~~~de~'d@=:tf'  ~de~' 

de       dR    dTr       dR    dfl 

3j3l       dir     dfil       dfl    ^(Bl 

^T  _          dfl_-[ 


'   de     a 
de 

' 


dR      dR      dR      dR 


also 


dR      dR     de       dR    dir  de 

^/o~  ==  ~^7~ '  ~^Q    I    5Z7  '  xa  »     and     ~o  =  1 , 


Sir 


1. 


'W 


dR      dR      dR 

=  ^r  + 


de     r  dTr' 


22 


In  a  similar  way  we  get  the  following  as  the  partials  of  R  with 
respect  to  a ,  a^ ,  and  a2  respectively : — 

a  (l-e2)  dfi__3a 
1  dR 


1       i/l  —  e2  dR         1  cos  i 

daz~       &!/«          G  de       kVa    i/l  —  e2  sin  i    &  ' 

Put  n  =  ^/a^,  and  substitute  these  values  in  (13),  then 


(14) 


da_^ 

di  ~na'~de' 


dt       no2  i/l  —  e2  sin  i    ^ 


d'jr  2 

^       no2  i/l  —  e2 

^e 


^  + 


(Zi 
5^ 


wa2        a?r 
-1 


-yT= 


l-l/l-e2  a^ 


—  e"  sin 


+ 


a€     -2 

dt  ~~  na 


+  \/i- 


1-i/l-e2 


de 


23 

VI. 

TRANSFORMATION  OF  EQUATIONS  EXPRESSING  THE  PERTURBA- 
TIONS AND  THE  VALUES  OF  THE  VARIATIONS. 

The  perturbations  can  be  expressed  in  another  form  by  the  fol- 
lowing substitutions: — 

Let  the  perturbing  force  which  m  exerts  upon  m  at  any  instant 
be  resolved  into  three  rectangular  components  as  follows  : — 

(1)  (w'/l  +  m) R'  is  the  component  along  the  radius  vector  of 
m1  reckoned  positive  away  from  the  sun. 

(2)  (m'/l  +  fn>)  &'  is  the  component  perpendicular  to  the  radius 
vector  and  in  the  plane  of  the  orbit,  positive  in  the  direction  of 
motion. 

(3)  (m'/l  +<m)W  is    the   component   perpendicular   to  the 
plane  of  the  orbit,  positive  northward. 

Hence  each  of  the  variations  of  R  along  the  coordinate  axes  will 
be  made  up  of  three  parts,  and  will  be  determined  by  the  equa- 
tions, 

m    dR  cos  +         cog 


m!        dx 


~  ^  =  R'  cos  R'Y  +  &  cos  S'Y  +  W  cos  TFT", 


m'        % 

.#'  cos  .#'£  +  #'  cos  tf'Z  +  IF'  cos  TF'Z. 


m'         dz 

The  values  of  the  cosines  can  be  obtained  from  the  following 
spherical  triangles : — 

Let  the  plane  of  JTJTbe  the  ecliptic  (the  X  axis  passing  through 
the  vernal  equinox)  and  T  the  point  where  it  is  cut  by  the  plane  of 
the  orbit  of  m;  OR ',  the  radius  vector  of  m,  then  the 

gives 

cos  R' X  —  cos  U  cos  u  -f  sin  U  sin  u  cos  (180  —  i) , 

=  cos  ft  cos  w  —  sin  ft  sin  u  cos  ^ . 


24 

Let  /S"  0  be  drawn  in  the  plane  of  the  orbit  perpendicular  to 
OR,  then  /  #OT  =  90  +  u,  and  AJTT#'  gives 

cos  S'X=  cos  fl  cos  (90°+w)  +  sin  O  sin  (  90°+w)  cos  (180  -  *)  , 
=  —  cos  H  sin  u        —  sin  fl  cos  u  cos  i  . 

Let  W  0  be  drawn  perpendicular  to  the  plane  of  the  orbit, 
then  /  TF'TX=  (90  -  »),  and  AXT  TF'  gives 

cos  TF'X  =  cos  n  cos  90°  +  sin  fl  sin  90  cos  (90  —  i), 
=  +  sin  U  sin  i  . 

On  substituting  these  cosines  in  our  first  equation  above,  we 
have 

-  ?  --  5—  =  It'  fcos  u  cos  U  —  sin  u  sin  fl  cos  £1 
m       dx 

4-  /S"  [—  sin  w  cos  O  —  cos  u  sin  H  cos  *]  4-  W  [sin  U  sin  *]  . 
In  like  manner  we  can  derive  the  equations 

-  -,  --  ^—  =  H'  fcos  u  sin  U  4-  sin  w  cos  U  cos  i  1 
m       dy 

4-  /S"  [—  sin  w  sin  ft  +  cos  w  cos  U  cos  £  ]  4-  W  [—  cos  U  sin  i~\. 
and 

-  ^—  •  -=—  =  j?T  sin  w  sin  i  1  4-  >S"  ["cos  u  sin  *  1  +  W  \  cos  i  1  . 
m          dz 

By  means  of  the  above  expressions  we  can  express  the  partials 
of  the  Perturbing  Function  (  R  )  contained  in  equations  (14)  in 
terms  of  the  components  R'  ',  $',  and  W.  As  an  example  we 
shall  find  the  value  of  dRjda: 

Now 


-  —  [—  —1   —  \d~y~  — 

~~fa[_dr  'da]  +  "^  \_~dr  'da 


— 

da 


25 
From  the  properties  of  an  ellipse,  r  =  a(l  —  e  cos  _Z?), 

dr      r         .  k  ... 

.  •  .  —  =  —  ,     also     n  =  -;  =  mean  daily  motion  , 
da      a  a* 

then 

jE'  —  e  sin  E  —  nt  -f  e  —  TT  , 

v         II  +  e       E 

tan2=>lr^tan^' 

and 


(16) 


cc  =  r  {[cosll  cos  w]  —  [sinfl  sinit  cos*]}, 
y  =  r  {  [  sin.fl  cos  w  ]  —  [  cos  U  sin  w  cos  i  ]  }  , 
z  =  r  {[sin  w  sin  i]}, 


whence  we  express  x,  y,  and  g  in  terms  of  the  constants  a,  a1?  a2, 
and  /3,  ^,  /32,  and  «;  likewise,  dx/dt,  dy/dt^smd  dz/dt.  By 
differentiating  (16)  we  have 


—  - 
dr 


=  fcos  H  cos  u  —  sin  H  sin  u  cos  £1  =  cos  R'  X, 


y 
-*?-=  [sin  fl  cos  ^  +  cos  fl  sin  u  cos  i~]  =  cos  jff'  J", 

^-  =  [sinwsin*],  =cos^'Z, 

and  substituting  in  (15) 


Now  we  substitute  in  this  equations  the  value  of  d  Rjdx  ,  d  Rjdy 
and  dRjdz  derived  in  the  beginning  of  this  article;  and  it  re- 
duces to 

dE      r  (    m'          \ 
da  ~~  a  \l  +m      )' 


26 

for  the  terms  containing  the  squares  of  the  cosines  and  their  prod- 
ucts can  be  reduced  by  the  formulae 

cos2  a  -f-  cos2  /3  +  cos2  7  =  1,      cos2  a  -f  cos2  /3'  -f  cos2  7'  =  1  , 
cos  a  cos  a!  +  cos  0  cos  /3'  +  cos  7  cos  7'=  0  , 

where  the  cosines  are  the  direction  cosines  of  two  lines. 
To  find  dRjdi  we  have 


<te~|    ,   f^»    ^1    ,   f 

di  \  +  [  dy  '&  J     L 


- 

dz    di 


And  from  (16)  we  get 


—  -  =  r  [sin  u  sin  O  sin  z  1 
^1 

G  y 

-~  =  /•  [—  sin  M  cos  U  sin  ^] 

dz 

—  -  =  r  [sin  w  cos  1  1 
di 

and  knowing  dfi/dx,  dfi/dy,  and  dRjdz^  dRjdi  reduces  to 
dE         m 


di     ~  1  -f  m 


r  W  sin  u . 


The  other  partial  differential  coefficients  can  be  found  in  a 
similar  manner.  If  these  coefficients  be  introduced  in  equations 
(14),  we  obtain  the  following  variations  of  the  elements  : — 


27 


(17) 


j^._^rsin^+^i 

1  +  m    VT^72L  >      r 


da        2m'  na3 

dt 


de         m'       na2  i/l  —  e2 

-ji  =  ^ — • —  • ^ [  sin  v2i 

dt      \  +  m  1 

+  (cos  v  +  cos  .#)  Sf] 
di         m  na 

dt=i^'yT=72[rGOSuWl 


»mi-dt  = 


-£  =  ^-    -•-    Ap--[-cos^' 
dt      1  -f  m  1 

4-  #'(1  +r)/psinv] 


or 

c?e  m'  _  _  „. 

-j-=-2an^—    -rJBf+  2sm2J-^  +2  sm2^    ,   , 
dt  1  +  m  2  eft  2   dt 

where  e  =  sin  <f> . 

VII. 

DR.  G.  W.  HILL'S  *  FIRST  MODIFICATION  OF  GAUSS'S  METHOD. 
If  the  orbits  do  not  intersect  each  of  these  differential  coefficients 
may  also  be  obtained  in  the  form  of  an  infinite  series  arising  from 
the  expansion  of  the  Perturbing  Function  to  terms  of  the  first  order 
with  respect  to  the  disturbing  forces.  Since  the  series  contains 
only  terms  of  the  form  A*£(iM  +  i'M')  in  which  A  is  a  con- 
stant and  i  and  i'  positive  integers,  it  follows  that  the  secular 

lOn  Gauss's  Method  of  Computing  Secular  Perturbations,  by  G.  W.  Hill, 
Astronomical  Paper  of  American  Ephemeris,  vol.  I. 


28 

portion  of  any  differential  coefficient  will  be  that  corresponding  to 
i  =  0  and  i'  =  0  .  If  we  consider,  for  example,  the  coefficient 
de/dt  we  will  have 


and  the  part  independent  of  Jf'  will  be 


the  part  of  this  series  independent  of  M  is  hence 


and  this  is  the  secular  part  of  the  perturbation. 

The  computation  of  the  secular  part  of  the  perturbations  is  thus 
reduced  to  evaluating  the  double  integrals, 

2lT~dMdM'. 
at 

etc.,  from  the  expressions  found  for  them  from  equations  (17)  of  the 
last  article.  The  integration  with  respect  of  M'  can  be  effected 
rigorously  in  terms  of  elliptic  integrals  of  the  first  and  second 
species,  but  that  with  regard  to  M  can  only  be  approximated  to 
by  mechanical  quadrature.  This  quadrature  is  more  accurate  if 
made  with  regard  to  E,  and  we  hence  transform  to  this  variable 
by  the  usual  formulae.  The  variable  M'  is  replaced  by  E'  also 
for  purposes  of  symmetry,  when  we  shall  have 


In  which  we  have  written  for  brevity 
_1_   C2"  ar 

etc. 


29 

Gauss's  method  of  effecting  the  integrations  (  b )  consists  in  re- 
placing the  variable  E'  by  a  new  variable  T  which  is  connected 
with  E'  and  ten  new  auxiliaries  by  the  following  equations : — 

^sin  E'  =  ft  -f  ff  sin  T  +  0"  cos  T, 
jVcos  E'  =  a  +  a'  sin  T  -f  a"  cos  T, 

If  =  7  -f  ry'  Sin  I7  +  7"  COS   I7. 

The  values  a ,  /3 ,  7  •  •  •  are  so  taken  that  the  coefficients  of  sin  T 
and  cos  T  vanish  in  the  expression 

A2  (7  +  7'  sin  T  -f  7"  cos  T)2 

[in  which  A  is  the  distance  between  the  two  bodies],  which  hence 
takes  the  form 


This  substitution  thus  finally  reduces  the  integrals  of  (b)  to  the  form 
a  sin2  T  -f  b  cos2  T 


r* 
Jo 


dT, 


in  which  «  and  5  are  independent  of  T  but  involve  G,  G',  and 
6r".  This  integral  can  readily  be  broken  up  into  elliptic  integrals 
of  the  first  and  second  kind  of  which  the  modulus 

2      G'  +  G" 

=  G  +  G" 

In  the  memoir  by  Dr.  G.  W.  Hill  the  steps  of  this  reduction 
will  be  found  given  in  detail  and  also  very  exact  tables  for  effect- 
ing the  computation.  These  quantities  of  the  tables  are  the  func- 
tions of  the  elliptic  integrals  met  with  in  evaluating  (  c  ).  They  are 
tabulated  to  the  argument  0  (  =  sin-1c  )  ,  and  are  published  to  eight 
decimals,  having  been  computed  to  ten. 

When  the  values  of  JS0,  SQ  and  WQ  have  been  found,  a  direct 
quadrature  of  (  a  )  will  be  resorted  to  to  effect  the  second  integra- 
tion. It  is  probable  that  no  accuracy  is  lost  by  our  inability  to 
exactly  integrate  these  expressions  since,  as  is  well  known,  the 
order  of  error  committed  cannot  in  any  of  the  coefficients  exceed  a 


30 


power  of  the  eccentricities  and  mutual  inclinations  of  the  orbits 
one  has,  than  the  number  of  parts  into  which  we  divide  the  orbit  of 
the  disturbed  body. 

VIII. 

COMPUTATION. 

The  following  is  an  application  of  Dr.  Hill's  method  to  the 
action  of  Jupiter  upon  Mars,  and  the  elements  employed  are  from 
Dr.  Hill's  "New  Theory  of  Jupiter  and  Saturn,"  pp.  192,  558. 


Mars. 

TT  =  333°  17'  51".74 

i  =      1    51      2  .24 

H  =    48    23    54  .59 

e=0.09326803 
n  =  689050".784 
log  a  =  0.1828971 
1 


m  = 


Jupiter. 

TT'  =  11°  54'  31".67 
'  it  =    1     18   42  .10 
fl'=98    56    19  .79 
e  =  0.04825511 
ri  =  109256".626 
log  a  =  0.7162374 

1 

m  = 


3093500.0  1047.879 

[Epoch  =  1850.0     G.M.T.] 

From  these  elements  the  preliminary  constants  become 


/=  1°26' 
n  =  149  47 
n'  =  188  22 
^T  =  321  24 
J5r=321  24 


6".38 

4  .37 

45  .43 

28  .27 
9.62 


log  k  =  9.9999971 
log  #  =  9.9998667 
log  c  =  8.7995614 
c  =  0.06303204 


If  the  orbit  of  Mars  be  divided  into  twelve  parts  with  regard  to 
the  eccentric  anomaly,  the  values  of  the  auxiliary  functions  cor- 
responding to  the  several  points  of  division  will  be  as  given  in  the 
following  tables.  A  rough  test  of  these  values  is  found  by  com- 
paring the  sums  of  the  functions  corresponding  respectively  to  the 
odd  and  even  points  of  division  of  the  orbit :  these  are  given  at  the 
foot  of  the  columns.  Sums  marked  thus  (*)  indicate  that  the 
corresponding  numbers  have  been  added  instead  of  their  logarithms 
as  given  by  the  points  of  division. 


31 


A  test  of  the  perturbations  in  the  plane  of  the  orbit  is  afforded 
by  the  condition  that,  since 


raa-} 

L  &  Joo 


0  ,     sin 


cos 


=  0  , 


this  residual  is  found  to  be  +  0.000,000,000,003,2. 

The  computation  has  not  been  duplicated,  but  various  checks  on 
the  accuracy  of  the  work  have  been  employed  as  the  computation 
progressed. 


E 

Logr 

V 

A 

log  B 

6 

0 

0.1403760 

o°  o'  o.'bo 

29.5201397 

0.9162454 

327°  64L59 

30 

0.1463201 

324724.62 

29.7305725 

0.9338723 

355  454.78 

60 

0.1621568 

644446.64 

29.8340289 

0.9418636 

22  20  51.41 

90 

0.1828971 

9521  5.91 

29.8101781 

0.9397444 

49  30  45.26 

120 

0.2026919 

124  31  47.16 

29.6691083 

0.9283109 

77  529.50 

150 

0.2166314 

152  34  23.40 

29.4449238 

0.9092471 

105  33  19.44 

180 

0.2216237 

180  0  0.00 

29.1903005 

0.8855869 

135  21  49.56 

210 

0.2166314 

207  25  36.60 

28.9697704 

0.8627418 

166  51  33.40 

240 

0.2026919 

235  28  12.84 

28.8461195 

0.8485520 

199  55  33.96 

270 

0.1828971 

264  38  54.09 

28.8598713 

0.8498206 

233  38  36.66 

300 

0.1621568 

295  15  13.36 

29.0110391 

0.8666375 

266  35  28.90 

330 

0.1463201 

327  12  35.38 

29.2554191 

0.8917695 

297  46  46.75 

8 

1.0916971 

900  0  0.00 

176.0707360 

5.3871961 

1028  25  54.92 

S' 

1.0916972 

1080  0  0.00 

176.0707353 

5.3871956 

1208  25  56.29 

E 

Logfir 

A      | 

G     \      G' 

0 

0.1016599 

27.008366 

2.448742 

27.0064625 

2.4695932 

30 

8.5335913 

27.006191 

2.661350 

27.0061438 

2.6618714 

60 

9.8433695 

27.007668 

2.763330 

27.0066000 

2.7737057 

90 

0.4413039 

27.011682 

2.735464 

27.0074721 

2.7765140 

120 

0.6339503 

27.014415 

2.591662 

27.0078882 

2.6581506 

150 

0.5856433 

27.013255 

2.368638 

27.0074658 

2.4330414 

180 

0.2641553 

27.009344 

2.117924 

27.0066063" 

2.1522696 

210 

9.2384099 

27.006392 

1.900346 

27.0061375 

1.9039682 

240 

9.5616845 

27.006977 

1.776111 

27.0064375 

1.7842143 

270 

0.3111659 

27.010167 

1.786673 

27.0071677 

1.8310715 

300 

0.5312995 

27.012575 

1.935433 

27.0075528 

2.0032709 

330 

0.4767380 

27.011664 

2.180724 

27.0071925 

2.2348548 

S 

11.8660187* 

162.059344 

13.633201 

162.0415473 

13.8412043 

S' 

11.8660198* 

162.059350 

13.633194 

162.0415794 

13.8413213 

32 


E 

G" 

6 

Log-R 

Log  I' 

Logfl 

o   /  // 

0 

0.0189480 

17  39  53.64 

0.03165480 

0.31498607 

0.22325501 

30 

0.0004748 

18  17  56.54 

0.03402429 

0.31811121 

0.22675996 

60 

0.009308F 

18  43  15.10 

0.03565127 

0.32025563 

0.22916456 

90 

0.0368400 

184858.26 

0.03602459 

0.32074753 

0.22971607 

120 

0.0599625 

18  28  29.55 

0.03469754 

0.31899872 

0.22775521 

150 

0.0586154 

17  39  45.34 

0.03164635 

0.31497491 

0.22324250 

180 

0.0316073 

16  30  39.87 

0.02757623 

0.30960115 

0.21721374 

210 

0.0033673 

15  24  39.12 

0.02396320 

0.30482483 

0.21185327 

240 

0.0075643 

145527.27 

0.02245002 

0.30282277 

0.20960581 

270 

0.0413999 

15  15  16.60 

0.02347167 

0.30417461 

0.21112339 

300 

0.0628166 

16  214.88 

0.02598814 

0.30750244 

0.21485859 

330 

0.0496602 

165332.60 

0.02889110 

0.31133795 

0.21916250 

8 

0.1902075 

10220  0.31 

0.17801800 

1.87416678 

1.32185292 

Sf 

0.1903576 

102  20  8.46 

0.17802120 

1.87417104 

1.32185769 

E 

•LogN 

LogP 

LogQ 

Log  V 

Ji 

0 

8.3476455 

5.799087ft 

7.1391282 

7.1387519 

27.0207829 

30 

8.3623563 

5.8175271 

7.1576461 

7.1576367 

27.0065656 

60 

8.3954328 

5.8524494 

7.1929779 

7.1927933 

27.0099909 

90 

8.436602T 

5.8931978 

7.2342423 

7.233512T 

27.0290204 

120 

8.4742977 

5.9283888 

7.2695991 

7.2684105 

27.0492441 

150 

8.4991682 

5.9492923 

7.2899853 

7.2888220 

27.0526533 

180 

8.5057537 

5.9513989 

7.2909894 

7.2903609 

27.0334422 

210 

8.4928482 

5.9346398 

7.2731849 

7.2731178 

27.0094522 

240 

8.4633467 

5.9029905 

7.2413630 

7.2412123 

27.0106059 

270 

8.4239466 

5.8638326 

7.2029257 

7.2021014 

27.0363166 

300 

8.3844575 

5.8269715 

7.1668219 

7.1655727 

27.0521291 

330 

8.3560126 

5.8027962 

7.1428979 

7.1419112 

27.0423359 

8 

50.570933S 

35.2612859 

43.3008794 

43.2971015 

162.1761951 

S/ 

50.570933$ 

35.2612856 

43.3008821 

43.2971011 

162.1763440 

33 


E 

J, 

J* 

LogJF3 

Log^3 

-Bo 

0° 

—0.20899495  , 

+0.30968486 

0.7664249 

9.059339671 

0.011523703 

30 

—0.03630764 

—0.02674098 

9.9823906 

8.3442271n 

0.011964426 

60 

+0.15284448 

—0.36413193 

0.637279771 

8.9312244 

0.012921849 

90 

+0.31348410 

—0.61207986 

0.936246971 

8.9588266 

0.014185451 

120 

+0.40064234 

—0.70414532 

1.032570171 

8.3275956n 

0.015433305 

150 

+0.38540725 

—0.61566127 

1.008416671 

9.169522071 

0.016299428 

180 

+0.26823871 

—0.37034154 

0.8476726n 

9.2218350n 

0.016504151 

210 

+0.08088125 

—0.03392182 

0.  3347999n 

8.766871771 

0.015978577 

240 

—0.12278036 

+0.30345688 

0.4964372 

8.8714541 

0.014896494 

270 

—0.28345025 

+0.55139873 

0.8711779 

9.0346961 

0.013593041 

300 

—0.35632065 

+0.64347029 

0.9812447 

8.2963250 

0.012429948 

330 

—0.32638729 

+0.55499846 

0.9539640 

8.9732022n 

0.011688529 

8 

+0.13362957 

—0.18200676 

—3.6043738* 

—0.123045628* 

0.083709450 

8' 

+0.13362742 

—0.18200674 

—3.6043746* 

—0.123045667* 

0.083709452 

E 

S0 

W0 

Sn 

Rn 

§ 

+0.00008004999 

+0.00041903914 

5.7629857 

30 

+0.00001088807 

—0.00003989400 

4.8906308 

7.6305418 

60 

—0.00007058128 

—0.00056153595 

5.686533971 

7.8866984 

90 

—0.00013852398 

—0.00104078831 

5.958627971 

7.9689460 

120 

—0.00017072125 

—0.00130819274 

6.029595971 

7.9232977 

150 

—0.00015776366 

—0.00121033849 

5.981375271 

7.6945110 

180 

—0.00010615130 

—0.00073761030 

5.804301671 

210 

—0.00003427562 

—0.00006864982 

5.3183538W 

7.685876771 

240 

+0.00003689362 

+0.00053477066 

5.3642593 

7.907922871 

270 

+0.00009184620 

+0.00088606326 

5.7801639 

7.950419571 

300 

+0.00012131643 

+0.00094343743 

5.9217628 

7.8698431n 

330 

+0.00011863616 

+0.00076351967 

5.9278969 

7.620409871 

S 

—0.00010919379 

—0.00071009176 

—0.000054778522* 

+0.0005847741* 

£' 

—0.00010919284 

—0.00071008769 

—0.000054774361* 

+0.0005847090* 

34 


E 

R0  sin  v  + 
S0(cos  v  +  cos  E  ) 

-*.*»,+fS.*n, 

Log  W0  sin  u. 

Log  W0  cos  u. 

\   a  cos2  <f>  / 

0° 

+0.0001601000 

—0.0115237021 

6.607402471 

6.0323895 

30 

+0.0064980855 

—0.0100466448 

5.4290192 

5.4698495w 

60 

+0.0116214735 

—0.0056380423 

6.0040176 

6.742246771 

90 

+0.0141365355 

+0.0010459980 

6.566602971 

6.989651471 

120 

+0.0128965787 

+0.0084589862 

6.997256971 

6.9298457n 

150 

+0.0077844290 

+0.0143154744 

7.0724419n 

6.419187071 

180 

+0.0002123026 

+0.0165041521 

6.8529748w 

6.2779619 

210 

—0.0072998748 

+0.0142155824 

5.705474971 

5.6648794 

240 

—0.0123115656 

+0.0083813594 

6.2543292 

6.702177471 

270 

—0.0135423545 

+0.0010841018 

6.1672230n 

6.9414069w 

300 

—0.0111295720 

—0.0055182011 

6.784158371 

6.857991871 

330 

—0.0061276125 

—0.0099498995 

6.861294271 

6.3702507n 

S 

+0.0014493172 

+0.0106645521 

—0.00243928405* 

—0.00233062987* 

sr 

+0.0014492082 

+0.0106646123 

—0.00243923736* 

—0.00233061181* 

E          --KO 

If  ra'  is  left  indefinite,  the 

resulting  values  of  the  dif- 

0°    —0.020897817 

ferential  coefficients  are  : 

Loe  coeff. 

30     —0.021996061 

[^1   =  +     165"71042m'        2.2193498 

60     —0.024638500 

Lat  Joo 

90     —0.028370902 

[dx~\ 

mf        4.1164137 

120     —0.032306045 

dt  Joo 

150     —0.035231951 
180     —0.036086917 

g]M=         268.8244 

1m'        2.4294687W 

210     —0.034538413 
240     —0.031182353 

f^l   =—  8712.3580 
L^Joo 

ro'        3.940135771 

270     —0.027186081 

300     —0.023700582 

['s]w=+13069-611 

m'        4.1162627 

330      -0.021488837 

S      —0.168812214 

L<^Joo==~19834'253 

mf        4.286327471 

S'     —0.168812245 

35 

If  the  above  value  of  m   be  employed,  we  get 
de 


00 


00 


0.15813891, 


12.476782, 


=-    0.25654148, 


=  -    83142788, 


oo 


00 


+  12.472445, 


=-18.450846. 


oo 


The  values  of  Newcomb  are  stated  on  page  378  of  his  "  Secular 
Variations  of  the  Orbits  of  the  Four  Inner  Planets."  Those  of 
Le  Verrier  are  found  in  the  Annales  de  1'Observatoire  de  Paris, 
Tome  II.,  page  101,  and  Tome  VI.,  page  189.  The  results  of  Le 
Verrier  have  been  reduced  to  the  above  value  of  m',  the  three  sets 
of  values  then  compare  as  follows : 


[ 


Results  of  Kesults  of  Method  of 

Le  Verrier.  Newcomb.  Gauss. 

^  I    =+    0.15810     +0.15818     +    0.15814, 


~ 
^  J 


e\^r-\    =+    1.16323     +1.16372     +    1.16328, 


oo 


~       =-    0.25640     -0.25655     -    0.25654, 

dt  Joo 


-    0.26873     -0.26850     -    0.26850, 

-18.45085. 


36 


BIOGRAPHICAL. 

The  writer  of  this  thesis  was  born  in  Baltimore,  Md.,  August 
11,  1872.  Prepared  at  Friends  Elementary  and  High  School, 
Baltimore,  Md.,  he  entered  the  Johns  Hopkins  University  in  1889, 
from  which  institution  he  received  the  A.B.  degree  in  1892. 
Graduate  student  in  Physics,  1892  to  February,  1893,  when  he 
accepted  a  position  in  the  U.  S.  Coast  and  Geodetic  Survey ;  1894— 
1895,  Principal  of  Martin  Academy,  Kennett  Square,  Pa. ;  1895, 
to  date  (including  absence  on  leave  1899-1900),  Professor  of 
Mathematics  in  Temple  College,  Philadelphia,  Pa.  While  teach- 
ing he  has  pursued  graduate  work  in  Astronomy,  Mathematics, 
and  Physics  at  the  University  of  Pennsylvania.  Also  pursued 
graduate  work  in  Physics  and  Mathematics  at  the  Johns  Hopkins 
University,  1899,  to  February,  1900. 


594329 


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